<h2>Problem 220</h2>
<div style="color:#666;font-size:80%;">06 December 2008</div><br />
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<p>Let <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" /> be the two-letter string &quot;Fa&quot;.  For n<img src='images/symbol_ge.gif' width='10' height='12' alt='&ge;' border='0' style='vertical-align:middle;' />1, derive <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>n</sub><img src="" style="display:none;" alt=")" /> from <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>n-1</sub><img src="" style="display:none;" alt=")" /> by the string-rewriting rules:</p>

<p style="margin-left:40px;">&quot;a&quot; <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> &quot;aRbFR&quot;<br />
&quot;b&quot; <img src='images/symbol_maps.gif' width='15' height='7' alt='&rarr;' border='0' style='vertical-align:middle;' /> &quot;LFaLb&quot;</p>

<p>Thus, <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>0</sub><img src="" style="display:none;" alt=")" /> = &quot;Fa&quot;, <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>1</sub><img src="" style="display:none;" alt=")" /> = &quot;FaRbFR&quot;, <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>2</sub><img src="" style="display:none;" alt=")" /> = &quot;FaRbFRRLFaLbFR&quot;, and so on.</p>

<p>These strings can be interpreted as instructions to a computer graphics program, with &quot;F&quot; meaning &quot;draw forward one unit&quot;, &quot;L&quot; meaning &quot;turn left 90 degrees&quot;, &quot;R&quot; meaning &quot;turn right 90 degrees&quot;, and &quot;a&quot; and &quot;b&quot; being ignored.  The initial position of the computer cursor is (0,0), pointing up towards (0,1).</p>

<p>Then <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>n</sub><img src="" style="display:none;" alt=")" /> is an exotic drawing known as the <i>Heighway Dragon</i> of order <i>n</i>.  For example, <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>10</sub><img src="" style="display:none;" alt=")" /> is shown below; counting each &quot;F&quot; as one step, the highlighted spot at (18,16) is the position reached after 500 steps.</p>

<div style="text-align:center;">
<img src="project/images/p_220.gif" alt="" /></div>

<p>What is the position of the cursor after 10<img src="" style="display:none;" alt="^(" /><sup>12</sup><img src="" style="display:none;" alt=")" /> steps in <b><i>D</i></b><img src="" style="display:none;" alt="_(" /><sub>50</sub><img src="" style="display:none;" alt=")" /> ?<br />
Give your answer in the form <i>x</i>,<i>y</i> with no spaces.</p>

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